Download Teaching the Standard Model in IB Physics by Debra Blake

Teaching the Standard Model in IB Physics
by Debra Blake Semmler, 2015 CTI Fellow
East Mecklenburg High School
This curriculum unit is recommended for:
High School Physics IB seniors (12)
Keywords: Physics, Particle Physics, Standard Model, International
Baccalaureate Physics
Teaching Standards: See Appendix 1 for teaching standards addressed in this
unit.
Synopsis: This curriculum unit will explore how to teach the Standard Model to
International Baccalaureate seniors. The aims of the International Baccalaureate Physics
course is to expose students to the most fundamental experimental science, which seeks
to explain the universe from the smallest particles to an understanding of the origins of
the universe. Yet more importantly, students will study the impact of physics on society,
the moral and ethical dilemmas, and the social economic and environmental impact of the
work of scientists in a global context. A unit of study in the IB physics core curriculum is
on the structure of matter; students will develop an understanding that matter consists of
six quarks and six leptons. Quarks were postulated on a completely mathematical basis in
order to explain patterns observed in properties of particles. Later large-scale
collaborative experimentation led to the discovery of the predicted fundamental particles.
Students will become proficient in their understanding of the standard model and how
conservation laws are applied in particle reactions.i
I plan to teach this unit during the coming year in to 15 students in Physics grades12.
I give permission for Charlotte Teachers Institute to publish my curriculum unit in print
and online. I understand that I will be credited as the author of my work.
Teaching the Standard Model in IB Physics
Debra B. Semmler
Introduction
What is the smallest particle of matter? This is a question that has been explored by
scientists since the beginning of time. The Greek philosophers were the first to ponder the
fundamental constitution of matter. They considered, that if you could cut matter into
smaller and smaller pieces you would end up with the smallest bits they called the atom,
which means that which cannot be cut. The search for the smallest particle started with
J.J. Thomson’s discovery of the electron in 1897 and was thought complete with
Rutherford’s discovery of the positive atomic nucleus and Niels Bohr’s orbital atomic
model at the beginning of the twentieth century. As experiments with the nucleus became
more sophisticated at higher and higher energies the number of small bits of matter
multiplied a hundred fold. Using Einstein’s famous E=mc2 formula a remarkable trade
off between energy and matter was able to transform energy of motion into new bits of
matter. The rules that control all matter transmutations are the most basic laws of nature,
in the conservation of energy, electric charge and momentum. After much work a model
describing the known particles in nature emerged, the Standard Model. It consists of only
twelve particles-six quarks and six leptons.ii This curriculum unit will explore how to
teach the Standard Model to International Baccalaureate seniors.
Classroom and School Environment
I teach at an urban, partial magnet high school with a total population of roughly 2,000
students, with approximately 900 students who are part of the International Baccalaureate
(IB) magnet. The school is comprised of approximately 52 % Africans American, 25 %
white, 16% Hispanics and 6 % Asian. More than 60% of the student population is on free
or reduced lunch. I will be using the curriculum unit in my IB physics III class; the
students in this course are on their third year in physics, having completed an honorslevel physics class as sophomores and as seniors they have completed science courses in
biology, earth and environmental science and honors chemistry. The IB physics course is
a college level physics curriculum divided over two years that includes a minimum of 40
hours of experimental work.
Rational
The aims of the International Baccalaureate Physics course is to expose students to the
most fundamental experimental science, which seeks to explain the universe from the
smallest particles to an understanding of the origins of the universe. Yet more
importantly, students will study the impact of physics on society, the moral and ethical
dilemmas, and the social economic and environmental impact of the work of scientists in
a global context. A unit of study in the IB physics core curriculum is on the structure of
matter; students will develop an understanding that matter consists of six quarks and six
leptons. Quarks were postulated on a completely mathematical basis in order to explain
patterns observed in properties of particles. Later large-scale collaborative
experimentation led to the discovery of the predicted fundamental particles. Students will
become proficient in their understanding of the standard model and how conservation
laws are applied in particle reactions.iii
Scientific Background
Discovery of the neutrino
There are three nuclear decay processes where the atomic mass remains constant and the
atomic number increases or decreases by one. They are all classified as beta decay, β. The
first is β– decay where a neutron in the nucleus changes into a proton with the emission of
an electron from the nucleus. The beta particle is the electron in this case. In β+ decay, a
proton in the nucleus changes into a neutron and a positron (the beta particle) is released
from the nucleus. With electron capture (EC) a proton in the nucleus changes into a
neutron by capturing an atomic electron, usually from the inner shell. The isotopes on the
neutron-rich side of the line of stability will tend to decay by β– emission, while those on
the proton-rich side will most likely decay by β+ or electron capture.iv
The following reaction represents the β– decay of 210Bi.
210
83Bi
→
210
84Po
+ e−
210
In this case the parent nucleus, 210
83Bi, decays into a daughter nucleus, 84Po, and an
–
electron (β particle) is emitted. Notice that the atomic mass number, A, is the same for
the parent and daughter nuclei. The β– decay has changed the balance of protons
compared to neutrons in the daughter nucleus. We can calculate the amount of energy
released in the decay by calculating the Q value for the reaction. The Q value is the
difference between the rest energy of the parent nucleus and the sum of the rest energies
of the products of the decay. In this case, the Q value is 1.16 MeV. The energy released
by the decay should be seen as the kinetic energy of the emitted electron. This predicts
that the kinetic energy of the electron should be the same each time this particular decay
takes place. However, experimentally, the energies of the electrons emitted in this beta
decay are observed to vary from zero to the maximum energy available Emax= Q. Figure 1
shows the energy spectrum of the electron emitted in this decay. The electrons is
observed to have a range of kinetic energies, not one unique value as would be expected
assuming energy, mass and momentum conservation.
Figure 1. Experimental energy spectrum for decay electrons from Bismuth 210v
The result is that in a particular β– decay event the electron carries away less
energy then the difference between the rest energy of the parent and daughter. This would
result in energy and momentum not being conserved. The solution to this problem was
first suggest by Wolfgang Pauli in 1930 when he proposed that a third particle was
emitted in β– decay that carried the energy, linear and angular momentum necessary to
conserve these quantities in each individual decay event. This new particle did not carry
electric charge and had a mass much les than that of an electron. In 1933 Enrico Fermi
developed a quantum theory of β– decay that incorporated this proposal and called the
new particle the neutrino (“little neutral one” in Italian.) In 1956, in an experiment
performed by Clyde Cowan and Frederick Reines, the first neutrinos were observed in the
laboratory. The decay of 210
83Bi now correctly written as
210
83Bi
→
210
84Po
+ e− + 𝜐
Where the 𝜐 represents the neutrino (in this case is it antineutrino).vi
Particle Zoo
Since the discovery of the electron neutrino, physicists have identified 13 fundamental
constituents of matter. Just three of these-the up quark (u), the down quark (d), and the
electron are needed to make protons and neutrons and to form atoms and molecules.
There are four fundamental forces that act in nature: the strong force, the weak force, the
electromagnetic force and the gravitational force. The strong force is responsible for
binding the particles in the nucleus (protons and neutron) together. The electromagnetic
force binds electrons to the atomic nuclei to form atoms. The weak force helps the decay
of heavy particles. The gravitational force acts between massive objects and has no roll
on the subatomic scale. Particles transmit forces through exchange particles. Each force
has its own exchange particle.vii



The gluon mediates the strong forces; it “glues” quarks together
The photon carries the electromagnetic force; it also transmits light
The W and Z bosons represent the weak force; they induce different types of
decays
Physicists suspect that the gravitational force may be associated with a boson particle
hypothetically called the graviton.
Antimatter
For every particle that physicists have discovered a corresponding antiparticle exists.
Antiparticles have the opposite properties of their corresponding particles. For example,
an antiproton has a negative electric charge while a proton is positively charged. In the
mid-1990’s physicists at CERN and Fermilab created the first anti-atoms, by carefully
adding a positron and an antiproton to form antihydrogen. Storing antimatter is difficult
task because as soon as an antiparticle and a particle meet they annihilate, disappearing in
a flash of energy.viii
The Standard Model
Physicist call the framework that describes the interaction between elementary building
blocks (quarks and leptons) and the force carriers (bosons) the Standard Model. Gravity
is not yet part of this framework. The biggest success of the Standard Model is the
unification of the electromagnetic and the weak forces into the electroweak force. This
milestone is comparable to the unification of the electric and magnetic forces into a single
theory by Maxwell in the 19th century.
Even though the Standard Model describes the interaction the sub atomic world, it
does not explain the whole picture. The theory does not incorporate the gravitational
force or explain questions such as what is dark matter. The Standard Model may only be
a small part of a bigger picture and new information from experiments will continue to
describe the functioning of the subatomic world.ix
Table 1 The Standard Model of matter x
Quarks
There are six quarks grouped into three generations. All have fractional electric charge
and distinct antiparticles. It is the quarks and antiquarks that bind together in multitude of
ways to form more than 200 particles in the particle zoo and accounting for the vast
majority of the visible mass of the universe. The bound states of the quarks and
antiquarks are call hadrons and there are two subgroups of hadrons. Three group quark
combinations are called baryons, which the proton and the neutron are the two most
common examples and when a quark and antiquark combine they form mesons.xi
Leptons
There are three generations of leptons each consisting of a charged lepton and its related
neutrino as shown in Table 1. The electron is the most familiar of the charged leptons and
the only one that is stable. Each charged lepton has a distinct antiparticle and for each
neutrino there is also an antineutrino although at this point in time it is possible that the
two are not distinct like the photon is its own antiparticle. Unlike the quarks there are not
lepton-lepton bound states. Leptons are also referred to having three flavors: electron,
muon, and tau.xii
Feynman diagrams
Feynman diagrams are space-time diagrams, that is, time (ct) versus space graphs, used to
describe interaction of quarks, leptons and the mediator of these interactions. They are
used to compute lifetimes and cross section for particle interaction events. The time (ct)
and space (x) axes are normally not drawn but time (ct) is positive upward and generally
drawn with time flowing horizontally toward the right as shown in Figure 2. Straight
lines with an arrow represent particles. A particle line whose arrow points backward in
time is interpreted as the corresponding antiparticle moving forward in time thus omitting
the use of over-bars for antiparticles. The lines are symbolic and do not represent the
particle trajectories. The diagrams show the interactions between particles and
electromagnetic phenomena. Interactions occur at the vertices and photons are shown as a
wiggly line.xiii
Figure 3 Feynman diagramxiv
Student activities
Appendix 2 has four student activities to aid them in understanding the nature and
experimental background of particle physics. The first is a worksheet reviewing how to
read a particle physics chart; the goal is to familiarize the students with the families of
matter and their relationship with forces. The second, is a tutorial into how to draw
Feynman diagrams adapted from quantum diaries website. After completion of this
activity students should be able to draw eight required Feynman diagrams. The final two
student activities are using actual experimental data. The first is to observe the tracks of
particles in a bubble chamber. The second is to use actual data from a proton-proton
collision to determine the mass of the top quark. These student activities are adapted from
Topics in Modern Physics Particles by Leon Lederman Fermi lab group. Also included
with student activities are teacher notes and answers to the student worksheet.
Appendix 1: Implementing Teaching Standards
The Structure of Matterxv
Predictions: Our present understanding of matter is called the Standard Model, consisting
of six quarks and six leptons. Quarks were postulated on a completely mathematical basis
in order to explain patterns in the properties of particles. It was much later that large-scale
collaborative experimentation led to the discovery of the predicted fundamental particles.
Student should understand








Quarks, leptons and their antiparticles.
Hadrons, baryons and mesons.
The conservation laws of charge, baryon number, lepton number and strangeness.
The nature and range of the strong nuclear force, weak nuclear force and
electromagnetic force
Exchange particles
Feynman diagrams
Confinement
The Higgs boson
Students should be able to:







Describe the Rutherford-Geiger-Marsden experiment that led to the discovery of
the nucleus
Applying conservation laws in particle reactions
Describing protons and neutrons in terms of quarks
Comparing the interaction strengths of the fundamental forces, including gravity
Describing the mediation of the fundamental forces through exchange particles
Sketching and interpreting simple Feynman diagrams
Describing why free quarks are not observed
Appendix 2 Student Activities and Handouts
THE STANDARD MODEL OF ELEMENTARY PARTICLES CHART THE
ULTIMATE PERIODIC TABLE
WHAT CAN WE LEARN FROM IT?xvi
The purpose of this activity is to familiarize you with the Standard Model of Elementary
Particles by studying the Standard Model of Elementary Particles Chart.
As you look at the Chart, you will see that it is divided into four sections. The two top
sections list a total of 24 elementary particles. You will need information from these two
sections to answer the following questions.
1. The two top sections are labeled ________________ and __________________ .
2. For the time being, let’s disregard the right hand side of the Chart and look at the
left side labeled Matter. This category is further divided into two groups of six
particles. These groups of particles are given the names _________________ and
_______________ .
3. List the flavors (names) of the six quarks.
_______________ , _______________ , _______________ ,
_______________ , _______________ , _______________
4. List the flavors (names) of the six leptons.
_______________ , _______________ , _______________ ,
_______________ , _______________ , _______________
5. The symbol for each quark is ____________________________ .
6. Using the Chart, write the symbols for the following particles: up quark ________ down quark _______ top quark _______ charm quark
_______
7. Look at the leptons on the Chart. Their symbols (except for the electron) are
Greek letters. Fill in the symbols below
Lepton
Symbol
Name of Greek letter
Since there are three different neutrinos, how do their symbols distinguish them from
one another? ______________________________
Write the symbol for an electron neutrino: ___________
8.
Given the list of particles below, circle the quarks. (Do this without looking at the
Chart if you can.)
up, neutrino, electron, down, tau, charm, strange
9. Using the legend in the lower right-hand corner of the Chart, write down the
charge and approximate mass of each of the following:
Particle
Charge
Mass
up
strange
top
electron
tau
Electron neutrino
10. Why can mass be measured in MeV/c2 ?
11. Using the Chart complete the following: _______________ have charges that are
integers, and _______________ have charges that are fractions.
12. Baryons are particles that are made from quarks. The most common baryons are the
neutron and the proton. Applying the law of conservation of charge (that is, no charge
can be created or destroyed), what is the minimum number of quarks that must be joined
to make up one baryon that has a charge of either +1, -1, or 0? _________
Show the proof of your answer here.
13. List three combinations of quarks that will give you a baryon with a charge of:
a) +1 b) -1 c) 0
EXAMPLE: A baryon composed of ccs has a charge of +2/3 + 2/3 - 1/3 = +1.
QUARK COMBINATIONS FOR:
+1 -1 0
________
________
________
14.
________
________
________
________
________
________
The quarks and leptons in column 1 of the Chart make up all the stable matter
such as protons and neutrons. (Neutrons are stable relative to other particles,
although they can decay.) Apply this information to write the quark configuration
for a proton and for a neutron.
proton _______________ neutron _______________
15.
Which leptons are found in Column (or Family) I? ______________ Do you think
these are stable too? __________________
Now let’s look at the right side of the Chart, marked Antimatter. Using the Chart, explain
how the antiparticles differ from the particles in:
a) Charge _________________________
b) Mass _________________________________
c) Symbol ___________________________
(Which particle is an exception?) _________________________
d) Color _____________________________________
An antibaryon is made from antiquarks.
EXAMPLE: The antiquark configuration for an
antiproton is uud since it has a charge of -1; strangeness = 0. Write the antiquark
configuration for the following:
Antineutron () Antisigma minus ()
Antiomega plus ( )
The Chart in the bottom left-hand corner lists the Gauge bosons. These are the carriers of
the four fundamental forces in nature. List the four forces and their carriers.
Force
Carrier
1. Which particles are charged? _______________
2. Which force has the shortest range? __________
3. Which force(s) is/are affected by the color? _______________
4. Which force is the weakest? ____________
5. Which force holds the quarks together to form baryons? __________________
Answer Key
1.
2.
3.
4.
5.
6.
7.
matter, antimatter
quarks, leptons
up, charm, top down, strange, bottom
electron, muon, tau electron neutrino, muon neutrino, tau neutrino
the first letter of its name
u, d, t, c
 , ,subscript, e
8. up, down, charm, strange
9. +2/3, 3 MeV/c2 -1/3, 115 MeV/c2 +2/3, 175,000 MeV/c2 -1, 0.511 MeV/c2
-1,
1784 MeV/c2
0, < 3 10-6 MeV/c2
10. mass-energy equivalence
11. leptons, quarks
12. 3; two quarks will have a total charge of +1/3, +4/3, or -2/3.
13. Some of the possible combinations are: +1 (ucd, utb, ucs); -1 (dsb, dds, ddb); 0
(uds, cdb, tsb)
14. proton = uud, neutron = udd
15. electron, electron neutrino, yes
16. strong, gluons
weak, W+, W-, Zo electromagnetic, photons gravity, gravitons
17. W+, W18. weak
19. strong
20. gravity
21. strong
How to draw Feynman Diagrams
Adapted from Quantum Diariesxvii
For now, think of this as a game. You’ll need a piece of paper and a pen/pencil. The rules
are as follows (read these carefully):
You can draw two kinds of lines, a straight line with an arrow or a wiggly line:
You can draw these pointing in any direction.
You may only connect these lines if you have two lines with arrows meeting a single
wiggly line.
Note that the orientation of the arrows is important! You must have exactly one arrow
going into the vertex and exactly one arrow coming out.
Your diagram should only contain connected pieces. That is every line must connect
to at least one vertex. There shouldn’t be any disconnected part of the diagram.
What does it all mean?
Now we get to some physics.
Each line in rule (1) is called a particle.
The vertex in rule (2) is called an interaction.
The rules above are an outline for a theory of particles and their interactions. We
called it QED, which is short for quantum electrodynamics.
The lines with arrows are matter particles (“fermions”).
The wiggly line is a force particle (“boson”), which, in this case, mediates
electromagnetic interactions: it is the photon.
The diagrams tell a story about how a set of particles interacts.
We read the diagrams from left to right, so if you have up-and-down lines you should
shift them a little so they slant in either direction. This left-to-right reading is important
since it determines our interpretation of the diagrams.
Matter particles with arrows pointing from left to right are electrons.
Matter particles with arrows pointing in the other direction are positrons (antimatter).
In fact, you can think about the arrow as pointing in the direction of the flow of electric
charge. As a summary, we our particle content is: e+ is a positron, e- is an electron, and
the gamma is a photon… think of a gamma ray.
From this we can make a few important remarks:
The interaction with a photon shown above secretly includes information about the
conservation of electric charge: for every arrow coming in, there must be an arrow
coming out.
But wait: we can also rotate the interaction so that it tells a different story. Here are a few
examples of the different ways one can interpret the single interaction (reading from left
to right):
These are to be interpreted as: (1) an electron emits a photon and keeps going, (2) a
positron absorbs a photon and keeps going, (3) an electron and positron annihilate into a
photon, (4) a photon spontaneously “pair produces” an electron and positron.
On the left side of a diagram we have “incoming particles,” these are the particles
that are about to crash into each other to do something interesting. For example, at the
LHC these ‘incoming particles’ are the quarks and gluons that live inside the accelerated
protons. On the right side of a diagram we have “outgoing particles,” these are the things
which are detected after an interesting interaction.
For the theory above, we can imagine an electron/positron. In these experiments
an electron and positron collide and the resulting outgoing particles are detected. In our
simple QED theory, what kinds of “experimental signatures” (outgoing particle
configurations) could they measure? (e.g. is it possible to have a signature of a single
electron with two positrons? Are there constraints on how many photons come out?)
So we see that the external lines correspond to incoming or outgoing particles. What
about the internal lines? These represent virtual particles that are never directly
observed. They are created quantum mechanically and disappear quantum mechanically,
serving only the purpose of allowing a given set of interactions to occur to allow the
incoming particles to turn into the outgoing particles. Here is an example where we have
a virtual photon mediating the interaction between an electron and a positron.
In the first diagram the electron and positron annihilate into a photon, which then
produces another electron-positron pair. In the second diagram an electron tosses a
photon to a nearby positron (without ever touching the positron). This creates a visual
description of the idea that force particles are just quantum object that mediate forces.
However, our theory treats force and matter particles on equal footing. We could draw
diagrams where there are photons in the external state and electrons are virtual as shown
below:
This is a process where light (the photon) and an electron bounce off each other and is
called Compton Scattering. (1) that the electron emits a photon and then scatters off of
the incoming photon, or (2) we can say that the incoming photon pair produced with the
resulting positron annihilating with the electron to form an outgoing photon:
Anyway, this is the basic idea of Feynman diagrams. They allow us to write down
what interactions are possible. There is a much more mathematical interpretation of these
diagrams that produces the mathematical expressions that predict the probability of these
interactions to occur, and so there is actually some rather complicated mathematics.
However, just like a work of art, it’s perfectly acceptable to appreciate these diagrams at
face value as diagrams of particle interactions.
You should be able to draw Feynman diagrams for the following interactions;
1. Electron scattering
2. Beta decay
3. Pion decay
4. Electron positron annihilation
5. Pair production
6. Muon decay
7. Quark interactions
8. Photon -photon scattering
BUBBLE CHAMBER TRACK ANALYSIS
In this October, 1973 photo, 300 GeV protons are incident in the 30-inch hydrogen
bubble chamber. The electron spirals are counter-clockwise. Therefore, clockwise
deflection indicates a positive charge.
1. At point A we have a proton-proton interaction, which produces a π+ and many other
particles.
2. The π+ moves to B, where it recoils from an elastic collision with a proton.
3. The recoiling π+ slows as it travels to C. At C it decays according to the reaction:
+  +
+ 
The lack of forward momentum beyond point C indicates μ the likelihood of a decay of a
particle (π+ ), which was essentially at rest.
4. The μ+ produced at point C travels to point D. At D it decays according to the reaction:
+ 
e+
+ e
+

2
Determining the mass of the top quark using E = mc. Analysis of DZero Data from
Fermi National Accelerator Laboratory Student Handout
Introduction: Today, you will use Einstein’s famous equation and actual experimental
data collected in 1995 from a special event that is two-dimensional rather than threedimensional to determine the mass of the top quark. The top quark is the most massive
quark ever discovered.
Procedure - Part One: You will be given a computer-generated plot of a collision between
a proton and an antiproton. You will need to determine the momentum of each bit of
debris that comes from the collision. Be sure to remember that momentum has direction.
The collision may be represented in a general way as shown below. Your plot will show
this “top signature” but may not show the debris going in the directions shown here.
While this event looks complex at first, it may be summarized by noting that a proton
and antiproton collide to create a top-antitop pair that exists for a very short time. Almost
immediately the very massive top and antitop decay into the constituents that are known
as their signature. These include four “jets” (large blasts of particles) that are the result of
decays of W bosons and some less massive quarks. It is important to note that one of the
jets will often contain a low energy or “soft” muon. Soft muons help identify jets as
bottom quark jets. In addition, a muon and a neutrino come out as debris from the
collision.
You will notice that there is no information given about the neutrino except the magenta
tower indicating its direction on the color plot. While scientists can predict with
confidence that it comes out of the collision, it cannot be detected very easily. Still, a
careful consideration of the momenta before the collision and after the collision may give
you a clue about how much momentum this particle has!
Procedure - Part Two: Make a momentum vector diagram to determine the momentum of
the muon neutrino. Be sure to remember that the total momentum of the system must be
zero, so any “missing” momentum must belong to the neutrino.
Question 1. What is the momentum of the missing neutrino?
Procedure - Part Three: It turns out that if you are careful about your choice of units, it is
possible to equate momentum and energy in a way that is similar to the way mass and
energy are related. Specifically, it may be shown that the momentum you measured
above is the same numerical value as the energy or mass of the particle. In other words,
E (in GeV) = p (in GeV/c) = m (in GeV/c2)
This shows then that the total energy that
came from the two top quarks that were formed is equal to the numerical sum of all the
momenta discovered in the collision.
Fill in all the momentum values from your color
plot in the table below. Finally, add the measured value for the neutrino that you just
Momentum Jet 1
Energy or
Mass
Jet 2
Jet 3
Jet 4
Muon
Soft Muon
Neutrino
determined at the end of this table.
Question 2. What do you determine the mass of the top quark to be?
Go to the website for the Calculate the Top Quark Mass activity:
xviii
http://ed.fnal.gov/samplers/ hsphys/activities/top_quark_index.shtml. Under “Data and
Graphics,” there are three views of one event called Run 92704 Event 14022. These are
computer-generated pictures that represent the event discussed previously. To help
visualize the event, you may wish to first look at the color plot labeled Event 92704
14022 R-Z View, which gives a perspective from the side of the detector. Next, look at
the plot called Event 92704 14022 End View. Here the event is viewed in only two
dimensions, as seen from the end of the detector. Finally, you may view the color plot
called Event 92704 14022 Lego, which shows how the debris could be mapped if the
End View were unwrapped starting at the “X” axis. Below are images from the web site
Background Information for the Teacher
Energy, Mass and Momentum Calculations
In order for your students to find the mass of the top quark, they need to understand that
their discovery of the missing momentum of the neutrino is crucial. This value gives
them all they need to find the mass of the top quark. You will need to supply them with
the following information.
A common relation in high-energy physics is the following: E2 - p2 = m2
The reason energy, momentum and mass are shown as equal is actually due to the
convention of choosing a system where the speed of light, c, is set equal to one. In this
case, where particles are traveling with speeds of almost c, E = mc2 becomes E = m and
p = mv becomes p = mc or p = m. This does change scale somewhat to be sure, but it
allows for a simpler conversion between energy, mass and momentum.
In our particular case, it follows that one should write energy and momentum in terms of
the mass of the top quark.
E2 - p2 = (2m )2 t
When one observes that the net momentum before the collision is the same as the
momentum after and that value is zero, we write: or, taking the square root of both sides,
E2 = (2m )2 t
E = 2mt
Because almost all of the energy of the collision is the result of top and antitop decay, we
simply add the energies of the four jets, the soft muon, the muon and the muon neutrino
before dividing by the two tops (actually a top and an antitop quark) to obtain the mass of
the most recently discovered quark.
Students will use the values they calculated for momentum (now as energy values) and
incorporate their new value for the missing neutrino (in bold print below) before adding
all the energies as scalars to find 2mt.
348.2 GeV/2 = 174.1 GeV, which is very close to the currently accepted value of about
175 GeV. As was indicated, the “missing momentum” may be found in a more careful
analysis of the event as well as a better understanding of the event.
This relatively simple procedure may be repeated by your students in events generated by
the Fermilab computers that were directed to simulate various collisions that would show
this type of event.
Conclusion: The final result of this exercise should be that the students have gained some
experience in using actual data to see how scientists analyze collisions. Further, your
students will begin to understand that the latest pieces in the Standard Model puzzle have
been assembled from the energy in the Fermilab collider. Mass does indeed come from
energy as Einstein predicted.
This activity will build on your class’s understanding of vector addition and depend upon
only a short particle physics explanation from the instructor. The goal of this activity is
simple. Your students will determine the mass of the top quark.
Part I: Calculations of Momenta of Products of the Collision
The momentum of each jet or particle was determined by computer and is printed on the
color End View plots. These numbers will be used in creating a vector diagram of the
debris that comes from the collision as students attempt to find the momentum of the
“undetectable” neutrino.
Explain to the students that this is an exercise in momentum conservation. They are to
determine the momentum of the “undetectable” neutrino by adding up the vectors with
directions shown on the diagram and magnitudes indicated by the numbers listed. The
result should be a value close to the same number the computer determines for each
event. These values for the actual event and several computer-simulated events, each of
which has an End View plot available at the website, are listed below for the teacher’s
reference.
The direction of each neutrino may be verified by examining the color plots. Please be
aware that the students will not all get the exact values given above nor will they get the
precise directions shown on the pictures. This is due to their selection of the directions of
the debris in the first place as well as effects introduced by problems similar to the one
noted below.
Still, after vector diagrams are drawn by various groups of students, a reasonable value
for the momentum of each of neutrino may be found.
An example of a possible vector diagram for event 14022 is shown below for the
teacher’s convenience.
It is important to realize that this only works if the debris has no motion in the direction
of the beams which we define as the z direction. The event takes place in a plane
perpendicular to the axis of the proton and antiproton.
Resources
Bibliography for Teachers and Students
Bibliography
Bardeen, Majorie. Topics in Modern Physics Teacher Resource Materials. Fermilab. Jan
1, 2006. https://ed.fnal.gov/samplers/hsphys/tmp10-06.pdf (accessed July 20, 2016).
Teacher resources from Fermilab on how to teach particle physics in high school.
CERN. NEA Cern: The standard Model of Particle Physics. 2016.
http://www.nea.org/tools/lessons/CERN-the-standard-model-of-p[article-physics.html
(accessed Nov 5, 2016).
Teacher resources from CERN on the standard Model of particle physics
CERN The Standard Model. Jan 1, 2016. https://home.cern/about/physics/standard-model
(accessed Nov 20, 2016).
Elert, Glen. The Physics Hypertextbook. 1998. http://physics.info (accessed 09 10, 2012).
On-line physics text book
Fermilab. Science of Matter, Energy, Space and Time. July 20, 20162016.
http://www.fnal.gov/pub/science/particle-physics-101/science-matter-energy-spacetime.html.
From Fermi lab a publication on how to teach particle physics in a high school setting.
G. J. Neary. Warwick Elementry Physics Particle Group. Nov 5, 2016.
http://www2.warwick.ac.uk/fac/sci/physics/research/epp/exp/detrd/amber/betaspectrum/
(accessed Nov 5, 2016). Used for diagrams of Beta decay events.
Gleiser, Marcelo. NPR Cosmos and Culture. March 28, 2012.
http://www.npr.org/sections/13.7/2012/03/28/149457512/the-smallest-bits-of-things-abrief-history-of-matter (accessed Oct 20, 2016).
A time line of particle physics discoveries.
Grannis, Bill Carithers and Paul. Discovery of Top Quark. Sept 1, 1995.
http://www.slac.stanford.edu/pubs/beamline/25/3/25-3-carithers.pdf (accessed Nov 5,
2016).
A paper on the discovery of the Top Quark.
International Baccalaureate Organization. General regualations: Diploma Programme
For students and their legal guardians. Jan 1, 2007.
http://www.ibo.org/documentlibrary/regs_ibworldschools/documents/DP_regs_en.pdf
(accessed July 30, 2013).
IB Physics curriculum documents
Pearson NCS. Tutor Vista Particle Physics Feynman Diagrams. Jan 1, 2016.
http://physics.tutorvista.com/modern-physics/feynman-diagrams.html (accessed Oct 20,
2016).
On-line tutorial on how to draw Feynman Diagrams
Quantum Diaries. Quantum Diaries How to Draw Feynman Diagrams. Feb 14, 2010.
http://www.quantumdiaries.org/2010/02/14/lets-draw-feynman-diagams/ (accessed Oct
20, 2016).
On-line tutorial on how to draw Feynman Diagrams.
Quigg, Chris. American Physics Society. May 1, 1996.
http://lutece.fnal.gov/Papers/PhysNews95.html (accessed Oct 20, 2016).
Tipler, Paul A., and Ralph A. Llewellyn. Modern Physics. New York, NY: W. H.
Freeman and Company, 2008.
A paper by Ralph A Llewellyn on how to teach modern physics to high school students
i
(International Baccalaureate Organization 2007)
(Gleiser 2012)
iii
(International Baccalaureate Organization 2007)
iv
(Tipler and Llewellyn 2008)
v
(G. J. Neary 2016)
vi
(Bardeen 2006)
vii
ibid
viii
ibid
ix
(CERN 2016)
x
(CERN The Standard Model 2016)
xi
(Bardeen 2006)
xii
ibid
xiii
ibid
xiv
(Pearson NCS 2016)
xv
(International Baccalaureate Organization 2007)
xvi
(CERN The Standard Model 2016)
xvii
(Quantum Diaries 2010)
xviii
http://ed.fnal.gov/samplers/ hsphys/activities/top_quark_index.shtml
ii